# Optimizing adiabatic quantum program compilation using a graph-theoretic framework

- 46 Downloads

## Abstract

Adiabatic quantum computing has evolved in recent years from a theoretical field into an immensely practical area, a change partially sparked by D-Wave System’s quantum annealing hardware. These multimillion-dollar quantum annealers offer the potential to solve optimization problems millions of times faster than classical heuristics, prompting researchers at Google, NASA and Lockheed Martin to study how these computers can be applied to complex real-world problems such as NASA rover missions. Unfortunately, compiling (embedding) an optimization problem into the annealing hardware is itself a difficult optimization problem and a major bottleneck currently preventing widespread adoption. Additionally, while finding a single embedding is difficult, no generalized method is known for tuning embeddings to use minimal hardware resources. To address these barriers, we introduce a graph-theoretic framework for developing structured embedding algorithms. Using this framework, we introduce a biclique virtual hardware layer to provide a simplified interface to the physical hardware. Additionally, we exploit bipartite structure in quantum programs using odd cycle transversal (OCT) decompositions. By coupling an OCT-based embedding algorithm with new, generalized reduction methods, we develop a new baseline for embedding a wide range of optimization problems into fault-free D-Wave annealing hardware. To encourage the reuse and extension of these techniques, we provide an implementation of the framework and embedding algorithms.

## Notes

### Acknowledgements

The authors would like to thank Steve Reinhardt from D-Wave Systems Inc. and the anonymous reviewers for feedback. This work is supported in part by the Gordon and Betty Moore Foundation’s Data-Driven Discovery Initiative through Grant GBMF4560 to Blair D. Sullivan, a National Defense Science and Engineering Graduate Fellowship and a fellowship by the National Space Grant College and Fellowship Program and the NC Space Grant Consortium to Timothy D. Goodrich. This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

## References

- 1.Agarwal, A., Charikar, M., Makarychev, K., Makarychev, Y.: \(O(\sqrt{\log {n}})\) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems. In: Proceedings of the Thirty-seventh Annual ACM Symposium on Theory of Computing, pp. 573–581. ACM (2005)Google Scholar
- 2.Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys.
**74**(1), 47 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar - 3.Batagelj, V., Zaversnik, M.: An \(O(m)\) algorithm for cores decomposition of networks. arXiv preprint cs/0310049 (2003)Google Scholar
- 4.Beasley, J.E.: OR-library: distributing test problems by electronic mail. J. Oper. Res. Soc.
**41**, 1069–1072 (1990)CrossRefGoogle Scholar - 5.Boothby, T., King, A.D., Roy, A.: Fast clique minor generation in Chimera qubit connectivity graphs. Quantum Inf. Process.
**15**(1), 495–508 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar - 6.Boros, E., Hammer, P.L.: Pseudo-boolean optimization. Discrete Appl. Math.
**123**(1), 155–225 (2002)MathSciNetCrossRefMATHGoogle Scholar - 7.Britt, K.A., Humble, T.S.: High-performance computing with quantum processing units. arXiv preprint arXiv:1511.04386 (2015)
- 8.Cai, J., Macready, W.G., Roy, A.: A practical heuristic for finding graph minors. arXiv preprint arXiv:1406.2741 (2014)
- 9.Choi, V.: Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Process.
**7**(5), 193–209 (2008)MathSciNetCrossRefMATHGoogle Scholar - 10.Choi, V.: Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design. Quantum Inf. Process.
**10**(3), 343–353 (2011)MathSciNetCrossRefMATHGoogle Scholar - 11.D-Wave Systems Inc.: SAPI 2.4. D-Wave Systems Inc., Burnaby (2016)Google Scholar
- 12.Denchev, V.S., Boixo, S., Isakov, S.V., Ding, N., Babbush, R., Smelyanskiy, V., Martinis, J., Neven, H.: What is the computational value of finite-range tunneling? Phys. Rev. X
**6**(3), 031,015 (2016)Google Scholar - 13.Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol.
**173**(2005)Google Scholar - 14.Eppstein, D.: Parallel recognition of series-parallel graphs. Inf. Comput.
**98**(1), 41–55 (1992)MathSciNetCrossRefMATHGoogle Scholar - 15.Erdos, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci
**5**(1), 17–60 (1960)MathSciNetMATHGoogle Scholar - 16.Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution. arXiv preprint arXiv:quant-ph/0001106 (2000)
- 17.Hagberg, A.A., Schult, D.A., Swart, P.J.: Exploring network structure, dynamics, and function using NetworkX. In: Proceedings of the 7th Python in Science Conference (SciPy2008), pp. 11–15. Pasadena, CA, USA (2008)Google Scholar
- 18.Halldórsson, M., Radhakrishnan, J.: Greed is good: approximating independent sets in sparse and bounded-degree graphs. In: Proceedings of the Twenty-sixth Annual ACM Symposium on Theory of Computing, pp. 439–448. ACM (1994)Google Scholar
- 19.Hamilton, K.E., Humble, T.S.: Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets. arXiv preprint arXiv:1612.07366 (2016)
- 20.Hüffner, F.: Algorithm engineering for optimal graph bipartization. In: International Workshop on Experimental and Efficient Algorithms, pp. 240–252. Springer (2005)Google Scholar
- 21.Humble, T.S., McCaskey, A.J., Bennink, R.S., Billings, J.J., D’Azevedo, E., Sullivan, B.D., Klymko, C.F., Seddiqi, H.: An integrated programming and development environment for adiabatic quantum optimization. Comput. Sci. Discov.
**7**(1), 015,006 (2014)CrossRefGoogle Scholar - 22.Humble, T.S., McCaskey, A.J., Schrock, J., Seddiqi, H., Britt, K.A., Imam, N.: Performance models for split-execution computing systems. In: IEEE International on Parallel and Distributed Processing Symposium Workshops, 2016, pp. 545–554. IEEE (2016)Google Scholar
- 23.Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E
**58**(5), 5355 (1998)ADSCrossRefGoogle Scholar - 24.Kassal, I., Whitfield, J.D., Perdomo-Ortiz, A., Yung, M.H., Aspuru-Guzik, A.: Simulating chemistry using quantum computers. Ann. Rev. Phys. Chem.
**62**, 185–207 (2011)ADSCrossRefGoogle Scholar - 25.King, J., Yarkoni, S., Raymond, J., Ozfidan, I., King, A.D., Nevisi, M.M., Hilton, J.P., McGeoch, C.C.: Quantum annealing amid local ruggedness and global frustration. arXiv preprint arXiv:1701.04579 (2017)
- 26.Klymko, C., Sullivan, B.D., Humble, T.S.: Adiabatic quantum programming: minor embedding with hard faults. Quantum Inf. Process.
**13**(3), 709–729 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar - 27.Leighton, F.T.: Introduction to Parallel Algorithms and Architectures: Arrays Trees Hypercubes. Elsevier, Amsterdam (2014)MATHGoogle Scholar
- 28.Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci.
**20**(2), 219–230 (1980)MathSciNetCrossRefMATHGoogle Scholar - 29.Lokshtanov, D., Saurabh, S., Sikdar, S.: Simpler parameterized algorithm for OCT. In: International Workshop on Combinatorial Algorithms, pp. 380–384. Springer (2009)Google Scholar
- 30.Lokshtanov, D., Saurabh, S., Wahlström, M.: Subexponential parameterized odd cycle transversal on planar graphs. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 18. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2012)Google Scholar
- 31.Lucas, A.: Ising formulations of many NP problems. Front. Phys.
**2**, 5 (2014)CrossRefGoogle Scholar - 32.Lund, C., Yannakakis, M.: The approximation of maximum subgraph problems. In: International Colloquium on Automata, Languages, and Programming, pp. 40–51. Springer (1993)Google Scholar
- 33.Neven, H., Rose, G., Macready, W.G.: Image recognition with an adiabatic quantum computer I. mapping to quadratic unconstrained binary optimization. arXiv preprint arXiv:0804.4457 (2008)
- 34.Reed, B., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett.
**32**(4), 299–301 (2004)MathSciNetCrossRefMATHGoogle Scholar - 35.Rieffel, E.G., Venturelli, D., O’Gorman, B., Do, M.B., Prystay, E.M., Smelyanskiy, V.N.: A case study in programming a quantum annealer for hard operational planning problems. Quantum Inf. Process.
**14**(1), 1–36 (2015)ADSCrossRefMATHGoogle Scholar - 36.Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Ser. B
**63**(1), 65–110 (1995)MathSciNetCrossRefMATHGoogle Scholar - 37.Schmidt, J.M.: A simple test on 2-vertex-and 2-edge-connectivity. Inf. Process. Lett.
**113**(7), 241–244 (2013)MathSciNetCrossRefMATHGoogle Scholar - 38.Venturelli, D., Mandrà, S., Knysh, S., O’Gorman, B., Biswas, R., Smelyanskiy, V.: Quantum optimization of fully connected spin glasses. Phys. Rev. X
**5**(3), 031,040 (2015)Google Scholar - 39.Wang, C., Jonckheere, E., Brun, T.: Ollivier–Ricci curvature and fast approximation to tree-width in embeddability of QUBO problems. In: 6th International Symposium on Communications, Control and Signal Processing (ISCCSP), 2014, pp. 598–601. IEEE (2014)Google Scholar